Double trouble def double_trouble(items, n): Suppose, if just for the sake of argument, that the following operation is repeated n times for the given list of items: remove the first element, and append that same element twice to the end of items. Which one of the items would be removed and copied in the last operation performed? Sure, this problem could be finger-quotes “solved” by actually performing that operation n times, but the point of this exercise is to come up with an analytical solution to compute the result much faster than actually going through that whole rigmarole. To gently nudge you towards thinking in symbolic and analytical solutions, the automated tester is designed so that anybody trying to brute force their way through this problem by performing all n operations one by one for real will run out of time and memory long before receiving the answer, as will the entire universe. To come up with this analytical solution, tabulate some small cases (you can implement the brute force function to compute these) and try to spot the pattern that generalizes to arbitrarily large values of n. You can again also check out Wolfram Alpha or similar systems capable of manipulation of symbolic formulas to simplify combinatorial and summing formulas. The reason to pay attention in the discrete math and combinatorics courses is learning to derive symbolic solutions to problems of this nature so that you don't have to brute force their answers in a time that would be prohibitively long to ever be feasible. We met this very same theme back when counting the number of cannonballs that make up a brass monkey pyramid.

Double trouble
def double_trouble(items, n):

Suppose, if just for the sake of argument, that the following operation is repeated n times for the given list of items: remove the first element, and append that same element twice to the end of items. Which one of the items would be removed and copied in the last operation performed?

Sure, this problem could be finger-quotes “solved” by actually performing that operation n times, but the point of this exercise is to come up with an analytical solution to compute the result much faster than actually going through that whole rigmarole. To gently nudge you towards thinking in
symbolic and analytical solutions, the automated tester is designed so that anybody trying to brute force their way through this problem by performing all n operations one by one for real will run out
of time and memory long before receiving the answer, as will the entire universe.

To come up with this analytical solution, tabulate some small cases (you can implement the brute force function to compute these) and try to spot the pattern that generalizes to arbitrarily large values of n. You can again also check out Wolfram Alpha or similar systems capable of manipulation
of symbolic formulas to simplify combinatorial and summing formulas.

The reason to pay attention in the discrete math and combinatorics courses is learning to derive symbolic solutions to problems of this nature so that you don't have to brute force their answers in a time that would be prohibitively long to ever be feasible. We met this very same theme back when counting the number of cannonballs that make up a brass monkey pyramid.